Layered Medium Green's Functions Research Articles

Logarithmic singularity subtraction for matrix‐friendly layered medium Green's functions

AbstractA subtraction procedure is proposed to evaluate the logarithmic singularity for matrix‐friendly layered medium Green's functions. In this subtraction procedure, an integral of the Bessel function combined with the exponential function and power function, where the sum of the power of and the order of the Bessel function is −1, is designed to fit the oscillation of the tail integral. After subtracting the designed integral, the integrand of the remainder decreases rapidly. Accordingly, the remainder can be calculated without finding any surface poles or steepest descent path, while the subtracted part can be calculated analytically. Several numerical examples are provided to reveal the feasibility and the accuracy of the proposed method.

A Hybrid Loop-Tree FEBI Method for Low-Frequency Well Logging of 3-D Structures in Layered Media

Electromagnetic simulation plays an essential role in formation conductivity determination by electromagnetic well-logging tools, especially when the nonvertical borehole and invasion zones are present in a layered earth. A hybrid finite-element boundary integral (FEBI) method is suitable for such well-logging simulations. The usage of layered medium Green's functions allows the simulation background to be a planar stratified medium to characterize the earth formation. However, conventional FEBI using the Rao-Wilton-Glisson (RWG) basis function produces inaccurate results due to the low-frequency breakdown of the boundary integral equation solution. The new contribution of this work is to apply the loop-tree (LT) basis functions in the FEBI method to reduce the numerical error at low frequencies. Such an LT-FEBI method can handle all specific requirements of well-logging simulations at low frequencies. A direct solver is applied to make it more efficient to obtain logging curves with multiple source locations. The LT-FEBI simulation results are validated in several numerical examples, including a challenging well-logging model with a deviated borehole and invasion zones in the Oklahoma formation having 28 layers.

Error-Controlled Static Layered-Medium Green’s Function Computation via hp-Adaptive Spectral Differential Equation Approximation Method

A numerically robust and computationally efficient approach for evaluating a planar layered substrate's static Green's function is developed based on the adaptive form of the spectral differential equation approximation method. The method uses a pth-order finite element method (FEM) solution of the 1-D ordinary differential equation governing the spectrum of the layered-media Green's function with spatial h-adaptive meshing. The resulting pole-residue form of the Green's function spectrum enables analytic evaluation of the pertinent Sommerfeld integrals providing O(h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> ) error control of the spatial layered-medium Green's function in near, intermediate, and far zones. The detailed error analysis is presented enabling automation of the 1-D FEM mesh refinement, which guarantees a prescribed accuracy of the solution depending on the distance between the source and observation locations. The method is well suited for computing Green's function databases used by method of moments capacitance and inductance extractors.

Efficient Electromagnetic Modeling of Multidomain Planar Layered Medium by Surface Integral Equation

A novel electromagnetic modeling method for multidomain planar layered medium (MD-PLM) by surface integral equation (SIE) is proposed. The MD-PLM refers to the whole space that is separated into multiple PLM volumes, e.g., the infinite-sized PLM background with inserted finite PLM objects or multilayered parallel-plate structures. In the conventional SIE method for this scenario, only the exterior domain is considered as PLM that is modeled by layered medium Green's functions (LMGFs), while the interior domain is treated by the multidomain homogeneous SIE. However, in this work, the surface equivalence formulation for this special scenario is reestablished by the LMGFs. Then, both the exterior and interior domains are considered as PLM that is modeled by the corresponding LMGFs and SIE in each subdomain. As a result, only the boundary of each sub-PLM domain is required to be discretized rather than the interfaces of each interior medium, which is different from the way in conventional SIE. Consequently, compared with the conventional SIEs, the proposed method can reduce a large quantity of unknowns and significantly increase the efficiency by reducing the memory consumption. By some numerical examples, the performance of this method is evaluated and compared with the conventional methods. The validity and efficiency of the newly proposed method are also well demonstrated in the examples.

Calderón Preconditioned Spectral-Element Spectral-Integral Method for Doubly Periodic Structures in Layered Media

An efficient and accurate spectral-element spectral-integral method is developed for rigorously modeling arbitrary inhomogeneous objects with doubly periodicity embedded in a layered medium background. It is an improvement over the traditional finite-element boundary-integral (BI) method. In the proposed approach, the domain decomposition between the interior spectral-element domain and the outer surface BI domain enables the BI domain to be solved by a far more efficient spectral integral method using a uniform mesh; without any loss of generality, it leaves the interior spectral-element domain with flexibility to model any inhomogeneous objects. For the BI domain, with a uniform mesh and rooftop basis functions, we can exploit the Toeplitz structure of the BI matrices with the fast Fourier transform algorithm and apply the Calderón preconditioner to achieve a high iteration convergence rate that will not slow down as the problem gets larger. With the periodic layered medium Green's functions, the layers without objects can be treated as background and the discretization is avoided. We show that this method is very promising for large-scale problems, because the CPU time and memory required with respect to the number of unknowns grow slowly. We validate this method with a commercial finite element solver and show its ability by solving a large-scale optical lithography problem.

Numerical Simulation and Analysis of Multicomponent Induction Logging Response in Anisotropic Formation

In this article, the response characteristics of multicomponent induction logging (MCIL) tool in anisotropic formation are analyzed. To solve the 1-D MCIL problems, we use a planar layered anisotropic medium (PLAM) Green's function (PLAMGF) method which is derived from the layered-medium Green's function (LMGF) and combined with MCIL excitation source and anisotropic formation environment, and then use it to investigate the equivalence of the MCIL tool responses in two models, i.e. the microscopic thin interbed formation and macroscopic anisotropic formation, and also analyze the effects of shoulder bed and formation dip on MCIL tools. In addition, in order to solve the complex 3-D problems, such as the effects of borehole, tool eccentricity and invasion on the tool response, we use the finite element method (FEM) based on a proposed new meshing scheme to simulate and analyze. This new meshing scheme includes two aspects: the use of cylindrical hexahedron element and the flexible meshing strategy combined with domain decomposition. Finally, some key detection performances of coaxial coils and coplanar coils of MCIL tool are studied. Numerical experiments are conducted to prove the effectiveness of our proposed method and provide the detailed results of the aforementioned aspects.

Efficient Computation of Green's Functions for Lossy Uniaxial Anisotropic Layered Media

AbstractThe efficient computation of Green's functions for uniaxial anisotropic layered media is investigated. Both field and mixed‐potential layered medium Green's functions (LMGFs) are discussed for different applications. An asymptotic subtraction (singularity extraction) and the weighted average method are combined for the acceleration of LMGF computation. A full collection of Sommerfeld and related identities used in the acceleration is summarized and generalized to uniaxial anisotropic medium for both field and mixed‐potential LMGFs. When lossy media are considered, the effective distance from the source to the observation point becomes a complex number depending on the anisotropic ratio of the horizontal to vertical media parameters. In addition, the evaluation of half‐line source potentials, which are introduced to accelerate the scalar potential correction term for vertical currents, is generalized to the complex domain by analytic continuation. Several numerical examples are used to demonstrate the efficiency and accuracy of the resulting algorithms.

Surface-Volume-Surface EFIE for Electromagnetic Analysis of 3-D Composite Dielectric Objects in Multilayered Media

The surface-volume-surface electric field integral equation (SVS-EFIE) is generalized for the case of scattering problems on the composite nonmagnetic dielectric objects situated in planar nonmagnetic layered medium. The piece-wise homogeneous regions of the scatterer can be arbitrarily positioned with respect to the layers of stratification. The SVS-EFIE being a class of single-source integral equations is formed by restricting the surface single-source electric field representation in each distinct region of the scatterer through the volume-EFIE (V-EFIE) enforced on the boundary of that region for only the tangential component of the total field. As a result, the SVS-EFIE utilizes only the electric field dyadic Green's functions. This allows for its cast into the mixed-potential form using classical Michalski-Zheng's formulation and method of moments (MoM) discretization featuring easily computable integrals with singularities no stronger than 1/R, R being the distance from the source to the observation point in such integrals. The matrices of MoM discretization are represented inhierarchical form (as H-matrices) enabling solution of the scattering problems in multilayered media with O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> log N) CPUtimeand memory complexities, where α is a geometry-dependent constant ranging from 1 to 1.5 depending on the shape of the scatterer. While the MoM surface and volume meshes discretizing the regions of the scatterer are constructed to ensure that no mesh element crosses interfaces between the layers, the clusters of both the surface and volume elements in their respective recursive partitionings in the process of H-matrix construction are allowed to span multiple layers of the medium. Upon computation of the layered medium Green's function kernels with the discrete complex image method allowing clusters of elements to cross dielectric interfaces between the layers is shown to preserve compressibility of the corresponding H-matrix blocks.

Frequency-Domain Reverse-Time Migration of Ground Penetrating Radar Based on Layered Medium Green's Functions

A frequency-domain reverse-time migration (RTM) algorithm based on the layered medium dyadic Green's function (DGF) is proposed for high-resolution and efficient subsurface imaging using ground penetrating radar (GPR). Different from the conventional RTM, which is performed by the finite-difference time domain (FDTD) method, both the forward and backward extrapolation wavefields in this frequency-domain RTM are computed by the multiplication between the frequency spectra of the excitation source or the recorded GPR data and the layered medium DGFs. The final image is reconstructed by the direct summation of all the wavefields for all the sampling frequencies in the frequency domain, instead of transforming them back to the time domain and processing them. Two-dimensional (2-D) or 3-D numerical and laboratory experiments show that the proposed frequency-domain RTM algorithm can yield almost the same imaging results as the conventional time domain RTM algorithm but requires less than two orders of magnitude in computational costs. The proposed GPR imaging method is verified to be competent for the fast 3-D imaging of potential larva rocks under the lunar surface, which can provide critical information for the regolith-drilling task of the Chang-E 5 lunar exploration mission of China. Real-time subsurface imaging by RTM could be fulfilled and be widely applied to engineering geophysics, such as urban utility detection, through-wall imaging, and ice monitoring by ground-coupled or air-borne GPR.

Fast and accurate simulation of electromagnetic telemetry in deviated and horizontal drilling

Electromagnetic (EM) telemetry systems are widely used for measurement-while-drilling (MWD), especially in unconventional drilling. In this paper, a numerical method for the simulation of EM telemetry systems in deviated and horizontal drilling is introduced. The underground formation is assumed to be horizontally layered media, and the long, metal drill string acts as an antenna that may be treated as a thin wire excited by a gap voltage source located near the drill bit. The numerical model employs the electric field integral equation (EFIE) and the method of moments (MoM) to obtain a linear system for the axial current distribution on the drill string, which is subdivided into several one-dimensional segments. By using the thin wire kernel, near-field interactions can be evaluated accurately and efficiently, while also reducing problem dimensionality and singularity order of the homogeneous medium contribution to the layered medium Green's function (LMGF). The latter typically takes the form of a dyadic spectral domain integral, the asymptotics of whose integrands may be estimated and combined with Kummer's method to accelerate its evaluation, with the asymptotic correction terms evaluated in closed or easily evaluated forms. Numerical results are presented to verify the fast and accurate performance of the proposed approach. Results for signal voltage receivability by a surface antenna are also presented and discussed.

A Hybrid Solver Based on Domain Decomposition Method for the Composite Scattering in Layered Medium

In the framework of domain decomposition method, we present a novel hybrid solver for accurate electromagnetic simulations of composite objects in layered medium (LM). This hybrid solver combines the surface integral equation method and the finite element method to effectively minimize the simulation domain, where the effect of inhomogeneous background is involved by the LM Green's functions, and the field transmission between neighboring subdomains is realized by the Riemann solver. Numerical results are given to validate the proposed solver.

Efficient and versatile surface integral approach to light scattering in stratified media.

Recent advances in nano-optics elicit a growing need for more efficient numerical treatments of light-matter interaction in stratified backgrounds. While being known for its many favorable properties, usage of the surface integral approach is hindered by numerical difficulties associated with layered-medium Green's functions. We present an efficient and robust implementation of this approach, addressing the limiting issues. The singularity extraction method is generalized to account for the occurring secondary-term singularities. The resulting scheme thus allows for arbitrary positioning of the scatterers. Further, the laborious matrix-filling process is dramatically accelerated through a simple and robustly-devised, spatial interpolation scheme, completely devoid of integral evaluations. The accuracy and versatility of the method are demonstrated by treating several representative plasmonic problems.

Estimation of Hydraulic Properties of a Sandy Soil Using Ground-Based Active and Passive Microwave Remote Sensing

In this paper, we experimentally analyzed the feasibility of estimating soil hydraulic properties from 1.4 GHz radiometer and 0.8-2.6 GHz ground-penetrating radar (GPR) data. Radiometer and GPR measurements were performed above a sand box, which was subjected to a series of vertical water content profiles in hydrostatic equilibrium with a water table located at different depths. A coherent radiative transfer model was used to simulate brightness temperatures measured with the radiometer. GPR data were modeled using full-wave layered medium Green's functions and an intrinsic antenna representation. These forward models were inverted to optimally match the corresponding passive and active microwave data. This allowed us to reconstruct the water content profiles, and thereby estimate the sand water retention curve described using the van Genuchten model. Uncertainty of the estimated hydraulic parameters was quantified using the Bayesian-based DREAM algorithm. For both radiometer and GPR methods, the results were in close agreement with in situ time-domain reflectometry (TDR) estimates. Compared with radiometer and TDR, much smaller confidence intervals were obtained for GPR, which was attributed to its relatively large bandwidth of operation, including frequencies smaller than 1.4 GHz. These results offer valuable insights into future potential and emerging challenges in the development of joint analyses of passive and active remote sensing data to retrieve effective soil hydraulic properties.

Calderón Preconditioned PMCHWT Equations for Analyzing Penetrable Objects in Layered Medium

We study Calderón preconditioners for analyzing electromagnetic scattering by penetrable objects in a layered medium. To account for the scattering effects of the multilayered background, the layered medium Green's function is adopted in the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) method. However, similar to the free-space case, the spectrum of the resulting equation is undesirable. This leads to a slow convergence of an iterative solver, especially when the geometry is densely meshed. To improve the convergence, a highly effective preconditioner is proposed. Different from its free-space counterpart, the preconditioning operator is constructed based on the Calderón identities for inhomogeneous medium. To reduce the relatively high construction cost of the preconditioning operator, several alternative simplified schemes are proposed and analyzed. Finally, the performances of different preconditioners are examined and compared carefully through different numerical examples. It is shown that the convergence of the PMCHWT system in a layered medium can be significantly improved by using the proposed Calderón preconditioners.

A Calderón Preconditioner for the Electric Field Integral Equation With Layered Medium Green's Function

A Calderón preconditioner is developed for the analysis of electromagnetic scattering of perfect electrically conducting (PEC) objects embedded in a layered medium. The electric field integral equation (EFIE) is formulated with the kernel of layered medium Green's function to account for the effects from the multilayered background. The Calderón projector is derived based on the general source-field relationship and the extinction theorem for inhomogeneous environment in electromagnetic theory. The Calderón identities can be naturally deduced based on this projector, which is then leveraged to precondition the EFIE with layered kernel. An alternative implementation is then proposed to make the implementation of the preconditioner as efficient as the one in free space. Different numerical examples are designed to show the performance of the preconditioner, where the objects are located in different positions with respect to the layered medium, or different types of excitation are adopted. It is shown that the proposed effective and robust preconditioner makes the EFIE system converge rapidly in all cases, independent of the discretization density.

Three-Dimensional Synthetic Aperture Radar Imaging Through Multilayered Walls

Most of existing through-the-wall radar imaging (TWRI) algorithms deal with a single layer homogeneous wall in two-dimensional scenario. In this communication we present a generalized three-dimensional (3-D) beamforming algorithm for the focused imaging of targets behind multilayered building walls. The far field layered medium Green's function is incorporated in the 3-D beamformer for the compensation of the wall effect. 3-D polarimetric imaging is exploited in TWRI for enhanced target identification and feature extraction as well as wall effect mitigation. Numerical results show that the proposed 3-D beamformer provides high quality focused images in various wall-target scenarios.

Graphene in Layered Medium Applications

ABSTRACTGraphene can be defined either as an infinitely thin material with an optical conductivity or as a thin layer with a finite thickness and an effective complex electrical permittivity. Layered medium Green's functions are calculated according to these two definitions for multilayered structures including a graphene layer. It is found that numerical methods show a good agreement especially in the far field. Moreover, finite thickness method requires significantly less computation time than the effective complex permittivity method. Thus, considering the computation time and accuracy, the finite thickness approach is a better alternative to analyze the electromagnetic wave propagation and scattering in multilayered media with a graphene layer. © 2013 Wiley Periodicals, Inc. Microwave Opt Technol Lett 55:2293–2296, 2013

Wave Propagation in Parallel Plate Metallic Waveguide With Finite Conductivity and Three Dimensional Roughness

We study the effects of random roughness on wave propagation in a parallel plate metallic waveguide with finite conductivity. The rough surface is three dimensional (3D) with roughness heights varying in both horizontal directions. Integral equations are obtained from the extinction theorem formulated with layered medium Green's function. The second order small perturbation method is then applied to solve the integral equations. A closed form expression for the coherent wave is derived, which is expressed in terms of a three-fold Sommerfeld type integral due to the waveguide structure. Approximate methods are applied to calculate the Sommerfeld integral. The results based on the approximations are shown to agree with that of direct integration. The coherent wave enhancement factors of absorption are computed. Results of the present 3D roughness case are compared with the previous results of two dimensional (2D) roughness. The results for waveguides are also compared with the results obtained for a plane wave incident on a metal surface with 3D roughness. Results are illustrated for 3D roughness with a variety of power spectra. It is shown that enhancement factors of 3D roughness are larger than for 2D roughness. We also show that enhancement factors for a waveguide are larger than that of the plane wave case. Comparisons are also made between the theoretical results and measurements.

A New Green's Function Formulation for Modeling Homogeneous Objects in Layered Medium

A new Green's function formulation is developed systematically for modeling general homogeneous (dielectric or magnetic) objects in a layered medium. The dyadic form of the Green's function is first derived based on the pilot vector potential approach. The matrix representation in the moment method implementation is then derived by applying integration by parts and vector identities. The line integral issue in the matrix representation is investigated, based on the continuity property of the propagation factor and the consistency of the primary term and the secondary term. The extinction theorem is then revisited in the inhomogeneous background and a surface integral equation for general homogeneous objects is set up. Different from the popular mixed potential integral equation formulation, this method avoids the artificial definition of scalar potential. The singularity of the matrix representation of the Green's function can be made as weak as possible. Several numerical results are demonstrated to validate the formulation developed in this paper. Finally, the duality principle of the layered medium Green's function is discussed in the appendix to make the formulation succinct.

Study on spontaneous emission in complex multilayered plasmonic system via surface integral equation approach with layered medium Green’s function

A rigorous surface integral equation approach is proposed to study the spontaneous emission of a quantum emitter embedded in a multilayered plasmonic structure with the presence of arbitrarily shaped metallic nanoscatterers. With the aid of the Fermi's golden rule, the spontaneous emission of the emitter can be calculated from the local density of states, which can be further expressed by the imaginary part of the dyadic Green's function of the whole electromagnetic system. To obtain this Green's function numerically, a surface integral equation is established taking into account the scattering from the metallic nanoscatterers. Particularly, the modeling of the planar multilayered structure is simplified by applying the layered medium Green's function to reduce the computational domain and hence the memory requirement. Regarding the evaluation of Sommerfeld integrals in the layered medium Green's function, the discrete complex image method is adopted to accelerate the evaluation process. This work offers an accurate and efficient simulation tool for analyzing complex multilayered plasmonic system, which is commonly encountered in the design of optical elements and devices.